A050176 - OEIS

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A050176

T(n,k) = M0(n+1,k,f(n,k)), where M0(p,q,r) is the number of upright paths from (0,0) to (1,0) to (p,p-q) that meet the line y = x-r and do not rise above it and f(n,k) is the least t such that M0(n+1,k,f) is not 0.

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 2, 3, 1, 1, 4, 5, 5, 4, 1, 1, 5, 9, 5, 9, 5, 1, 1, 6, 14, 14, 14, 14, 6, 1, 1, 7, 20, 28, 14, 28, 20, 7, 1, 1, 8, 27, 48, 42, 42, 48, 27, 8, 1, 1, 9, 35, 75, 90, 42, 90, 75, 35, 9, 1, 1, 10, 44, 110, 165, 132, 132, 165, 110, 44, 10, 1

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OFFSET

1,8

COMMENTS

Let V = (e(1),...,e(n)) consist of q 1's, including e(1) = 1 and p-q 0's; let V(h) = (e(1),...,e(h)) and m(h) = (#1's in V(h)) - (#0's in V(h)) for h = 1,...,n. Then M0(p,q,r) = number of V having r = max{m(h)}.

f(n,k) = -1 if 0 <= k <= [(n-1)/2], else f(n,k) = 2*k-n.

LINKS

Table of n, a(n) for n=1..78.

Bruce E. Sagan and Joshua P. Swanson, q-Stirling numbers in type B, arXiv:2205.14078 [math.CO], 2022.

EXAMPLE

Rows:

1, 1;

1, 1, 1;

1, 2, 2, 1;

1, 3, 2, 3, 1;

1, 4, 5, 5, 4, 1;

1, 5, 9, 5, 9, 5, 1;